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A222403 Triangle read by rows: left and right edges are A000217, interior entries are filled in using the Pascal triangle rule.

%I A222403 #19 Apr 04 2018 15:44:27
%S A222403 0,1,1,3,2,3,6,5,5,6,10,11,10,11,10,15,21,21,21,21,15,21,36,42,42,42,
%T A222403 36,21,28,57,78,84,84,78,57,28,36,85,135,162,168,162,135,85,36,45,121,
%U A222403 220,297,330,330,297,220,121,45,55,166,341,517,627,660,627,517,341,166,55
%N A222403 Triangle read by rows: left and right edges are A000217, interior entries are filled in using the Pascal triangle rule.
%C A222403 In general, if the sequence defining the left and right edges is [a_0, a_1, ...], the row sums [s_0, s_1, ...] are given by s_0=a_0 and, for n>0,
%C A222403 s_n = 2a_n + Sum_{i=1..n-1} 2^(n-i) a_i.
%C A222403 Conversely, given the rows sums [s_0, s_1, ...], the edge sequence is [a_0, a_1, ...] where a_0=s_0 and, for n>0, a_n = (s_n - Sum_{i=1..n-1} s_i)/2.
%H A222403 Robert Israel, <a href="/A222403/b222403.txt">Table of n, a(n) for n = 0..10010</a>
%F A222403 G.f. as triangle: (1+x-4*x*y+x*y^2+x^2*y^2)*y/((1-y)^2*(-x*y+1)^2*(-x*y-y+1)). - _Robert Israel_, Apr 04 2018
%e A222403 Triangle begins:
%e A222403 0
%e A222403 1, 1
%e A222403 3, 2, 3
%e A222403 6, 5, 5, 6
%e A222403 10, 11, 10, 11, 10
%e A222403 15, 21, 21, 21, 21, 15
%e A222403 21, 36, 42, 42, 42, 36, 21
%e A222403 28, 57, 78, 84, 84, 78, 57, 28
%e A222403 ...
%p A222403 d:=[seq(n*(n+1)/2,n=0..14)];
%p A222403 f:=proc(d) local T,M,n,i;
%p A222403 M:=nops(d);
%p A222403 T:=Array(0..M-1,0..M-1);
%p A222403 for n from 0 to M-1 do T[n,0]:=d[n+1]; T[n,n]:=d[n+1]; od:
%p A222403 for n from 2 to M-1 do
%p A222403 for i from 1 to n-1 do T[n,i]:=T[n-1,i-1]+T[n-1,i]; od: od:
%p A222403 lprint("triangle:");
%p A222403 for n from 0 to M-1 do lprint(seq(T[n,i],i=0..n)); od:
%p A222403 lprint("row sums:");
%p A222403 lprint([seq( add(T[i,j],j=0..i), i=0..M-1)]);
%p A222403 end;
%p A222403 f(d);
%t A222403 t[n_, n_] := n*(n+1)/2; t[n_, 0] := n*(n+1)/2; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 20 2014 *)
%Y A222403 Other triangles of this type: A007318, A051666, A134634, A222404, A222405.
%Y A222403 Cf. A000217.
%Y A222403 Row sums are A005803.
%K A222403 nonn,tabl
%O A222403 0,4
%A A222403 _N. J. A. Sloane_, Feb 18 2013